Orthogonal projection of vectors

Question

How do I find the orthogonal projection of the vector (2, -3, 1) onto the space spanned by the vectors v1=(1, 2, 0) and v2=(1, -1, 1), and then present it as a linear combination of v1 and v2?

Answer 1

Given u = (2, -3, 1), v₁=(1, 2, 0) and v₂=(1, -1, 1), You want to find Proj_{v1, v2} (u), which can be found by formula(1)

Proj_{v1, v2} (u) = (u * v₁ ) * v₁/ ||v₁||² + (u * v₂ ) * v₂/ ||v₁||² -----(1) projection formula

= (2 * 1 + (-3) * 2 + 1 * 0 ) * (1, 2, 0) / || 1² + 2² +0²|| + (2 * 1 + (-3) * (-1) + 1 * 1 ) * (1, -1, 1) / || 1² + (-1)² +1²|| -----(2) plug in number

= -4/5 * (1, 2, 0) + 2 * (1, -1, 1) -----(3) format

= (6/5, -18/5, 2) -----(4) result

To present projection in linear combination, we can refer to step 3,

Proj_{v1, v2} (u) = -4/5 * v₁ + 2 * v₂

Hope you find the anwser helpful :)